Internal hedging of intermittent renewable power generation and optimal portfolio selection

Abstract

This paper introduces a scheme for hedging and managing production costs of a risky generation portfolio, initially assumed to be dispatchable, to which intermittent electricity generation from non-dispatchable renewable sources like wind is further added. The proposed hedging mechanism is based on fixing the total production level in advance, then compensating any unpredictable non-dispatchable production with a matching reduction of the dispatchable fossil fuel production. This means making no recourse to short term techniques like financial hedging or storage, in a way fully internal to the portfolio itself. Optimization is obtained in the frame of the stochastic LCOE theory, in which fuel costs and $\hbox {CO}_2$ prices are included as uncertainty sources besides intermittency, and in which long term production cost risk, proxied either by LCOE standard deviation and LCOE CVaR Deviation, is minimized. Closed form solutions for optimal hedging strategies under both risk measures are provided. Main economic consequences are discussed. For example, this scheme can be seen as a method for optimally including intermittent renewable sources in an otherwise dispatchable generation portfolio under a long term capacity expansion perspective. Moreover, within this hedging scheme, (1) production cost risk is reduced and optimized as a consequence of the substitution of the dispatchable fossil fuel generation with non-dispatchable $\hbox {CO}_2$ free generation, and (2) generation costs can be reduced if the intermittent generation can be partially predicted.

Proof of Eqs. (3.4) and (3.5)

Let us denote by D a generic deviation measure, like standard deviation or CVaRD. From Eq. (2.20), the cost risk of the hedged portfolio $D \big (P^\text {LC,w}_h(\xi )\big )$, can be expressed as⁶

$$\begin{aligned} D\big (P^\text {LC,w}_h(\xi )\big )= D\big (\big [{\bar{w}}^\text {ga}-h \gamma {\bar{w}}^\text {wi}\big ]P^\text {LC,ga}(\xi )+ \big [{\bar{w}}^\text {co}-(1-h) \gamma {\bar{w}}^\text {wi}\big ]P^\text {LC,co}(\xi )\big ). \end{aligned}$$

(A.1)

Since the coefficients of $P^\text {LC,ga}$ and $P^\text {LC,co}$ do not sum to 1, we can rearrange Eq. (A.1) in the following way

$$\begin{aligned} \begin{aligned} D\big (P^\text {LC,w}_h(\xi )\big )=\,&\frac{1-\gamma w^\text {wi}}{1+(1-\gamma )w^\text {wi}} \\&\times D\left( \left[ \frac{w^\text {ga}-h \gamma w^\text {wi}}{1-\gamma w^\text {wi}}\right] P^\text {LC,ga}(\xi )\right. \\&\left. + \bigg [\frac{w^\text {co}-(1-h) \gamma w^\text {wi}}{1-\gamma w^\text {wi}}\bigg ]P^\text {LC,co}(\xi )\right) , \end{aligned} \end{aligned}$$

(A.2)

in which Eqs. (2.18) and (2.19) were used. Since the coefficients of $P^\text {LC,ga}$ and $P^\text {LC,co}$ within the D operator sum to 1, in the case of the standard deviation the minimum risk hedging strategy can now be obtained by solving in h the equation

$$\begin{aligned} \frac{w^\text {ga}-h \gamma w^\text {wi}}{1-\gamma w^\text {wi}}=w^\text {ga}_{\mathrm{mvp}}. \end{aligned}$$

(A.3)

In the case of CVaRD the minimum risk hedging strategy can be instead obtained by solving in h the equation

$$\begin{aligned} \frac{w^\text {ga}-h \gamma w^\text {wi}}{1-\gamma w^\text {wi}}=w^\text {ga}_{\mathrm{mcp}}. \end{aligned}$$

(A.4)

Equations (3.4) and (3.5) follow.